Types directed by constants

Let T be a complete, countable, first-order theory having infinite models. We introduce types directed by constants, and prove that their presence in a model of T guaranties the maximal number of non-isomorphic countable models : I(א0, T ) = 20 . © 2009 Elsevier B.V. All rights reserved. Shelah in [6] proved that a countable theory with an infinite, definable, linear order and Skolem functions has 20 nonisomorphic countable models. He reduced the general case to the cases T1 = Th(ω,<, . . .) and T2 = Th(ω + ω, <, . . .) (where ω is reversely ordered ω, and is here attached on top of ω). In the first case he proved that an arbitrary countable, complete linear order can be ‘coded’ in a model of T1 by a chain of certain end extensions (using Rubin’s proof, see also [5]). In the second case, assuming thatω is not definable, Shelah first defined when two elements in a model of T2 are ‘near’, then showed that the model can be decomposed into convex, closed under nearness components and, finally, he showed that an arbitrary linear order with successors and predecessors can be coded by the order of components. Since there are 20 non-isomorphic countable, complete, linear orders with successors and predecessors, the conclusion followed. In this paper we are interested in coding linear orders in partially ordered structures (M,≤, . . .) which are somehow similar to the above. We fix an infinite subset C ⊆ dcl(∅) (playing the role of ω in a saturated model of T1 or T2) and show that an arbitrary linear order can be coded in a model of (a slight modification of) T = Th(M,≤, . . .), provided that T is small (|S(T )| = א0) and: (1) {x ∈ C | c ≤ x} is a co-finite subset of C for all c ∈ C; and (2) C is an initial part orM: c ∈ C andm ≤ c implym ∈ C. The slight modification is in that wemay need to absorb a (single) parameter into the language and to shrink C if necessary. Condition (1) here is quite strong, it is stronger than: (C,≤) is a directed, well partial ordering of height ω. We will see that its model-theoretic version, condition (D1) from Definition 2.1, describes it as a ‘generic linearity’, which seems to be a natural assumption needed for coding linear orders. To describe our proof assume that (M,≤, . . .) is saturated and that C ⊂ dcl(∅) satisfies the above two conditions. C determines a (possibly incomplete) 1-type pC(x) which is, as a subset of S1(∅), the set of all accumulation points of {tp(c) | c ∈ C}. We will work exclusively inside C ∪ pC(M), which turns out to be an initial segment of (M,≤) with E-mail address: [email protected]. 0168-0072/$ – see front matter© 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.apal.2009.12.002 P. Tanović / Annals of Pure and Applied Logic 161 (2010) 944–955 945 C < pC(M) (this is why we will call pC a C-directed type). The proof is based on the redefined nearness relation and the related C-independence. For a, b ∈ pC(M), we will consider a to be near b if a ≤ b and tp(a/b) is not finitely satisfiable in C (in our terminology: tp(a/b) is not a C-type); (a1, . . . , an, . . .) is C-independent if each tp(ak+1/a1a2 . . . ak) is a C-type (non-algebraic and finitely satisfiable in C). Assuming smallness, we will prove that our T can be slightly modified (and C replaced by a subset) so that the resulting pC(x) becomes complete and so called strongly C-directed. Roughly speaking, if the type is not strongly directed (which is the case in Shelah’s T1), wewill add a parameter tomake it look like T2, and thenwe will be able to get a strongly directed type. Thus we will reduce the general case to the T2-like case. In the case of a strongly directed type wewill show that the equivalence relation generated by our nearness is particularly well behaved: it will turn out that being in distinct classes is the same as being C-independent, and that the (complete) type of a C-independent set is determined by its<-type; in other words, if we factor out the equivalence, the only structure on the quotient is the one induced by<. Finally, we will show that any linear order can be coded in this way; thus, even in the case of Shelah’s T2, our coding is finer than his original one. Theorem 1. Suppose that T = Th(M,≤, . . .) is small, pC(x) ∈ S1(∅) is strongly (C,≤)-directed, and M |= T . (a) If I ⊂ pC(M) is a maximal C-independent set, then (I, <) is a linear order whose isomorphism type does not depend on the particular choice of I. (b) For every linear order there is N |= T such that the order type of a maximal C-independent subset of pC(N) is isomorphic to it. Corollary 1. If T is a complete, countable theory having a type over a finite domain which is directed by constants, then I(א0, T ) = 20 . The original motivation for my work comes from Anand Pillay’s work on elementary extensions of first-order structures in his Ph.D. thesis; see also [1–4]. There he proves that an arbitrary countable first-order structure has at least four countable elementary extensions which are non-isomorphic under isomorphisms fixing the ground structure point-by-point, and conjectures that the number must be infinite; in other words, he conjectures: If T is the elementary diagram of a countable model then I(א0, T ) ≥ א0. This article contains one main ingredient of the proof, Corollary 1 above. The other, a dichotomy theorem for minimal structures (every definable subset is either finite or co-finite), is contained in [9]. It asserts that either Sem (semi-isolation, defined below) is a pregeometry operator on the whole monster model, or there is a type (over a finite domain) directed by constants around. The two quickly produce the proof of the conjecture, as described in [9]. The paper is organized as follows: Section 1 contains a review of definitions and facts used later. In Section 2 we define C-directed types and prove that the definition is equivalent to the one sketched above. In Section 3 we prove that every definable subset of C ∪ pC(M) contains a minimal element, which will compensate the absence of Skolem functions. For the rest of the paper smallness of T will be assumed, and used essentially in the proofs. In Section 4 we consider intervals [a, b]where tp(a/b) is a C-type as ‘large’ ones, and prove that a ‘small’ interval cannot contain a large one; here, for a ≤ b, [a, b] is small means also that a is near b. In Section 5 strongly directed types are defined, and it is described how they can be obtained from directed types. In the remaining sections we focus on the locus of a strongly directed type. In Section 6 we prove first, that the equivalence relation induced by the nearness is the semi-isolation (x ∈ Sem(y)), then that the quotient set is linearly ordered, and then the uniqueness of the type of a C-sequence (of fixed length). The remaining needed fact for the proof of Theorem 1 is the degeneracy of C-independence: every pairwise C-independent set is C-independent; it also implies that the only structure on the quotient is the one induced by≤. Degeneracy is proved in the last section, where Theorem 1 is proved, too.